That's about correct. The color "Omaha Orange" was included in the Federal ambulance specifications, which I believe date originally from 1973 (+- a year). In an effort to standardize the appearance of ambulances, they were all to be painted white with an Omaha Orange stripe.
I believe the color was selected based on safety considerations and the fact that it didn't conflict with fire apparatus colors (red, lime green at the time, etc.).
The Federal specifications were not mandates, they were recommendations. However, if an ambulance was purchased with Federal funds, it was required to meet the Federal specs. At the time, "block grants" were available for the establishment and improvement of EMS systems, which included Federal money for the purchase of modern ambulances. So, if your service used block grant money to buy an ambulance, it had to meet the Federal specs, including the Omaha Orange stripe. And there were a lot of ambulances bought with block grant money at the time. The block grants ended with the end of the Carter administration.
Some states mandated ambulances meet the specs to receive a state license, no matter how you bought it. Many other agencies bought ambulances built to the Federal specs, even with their own money, as this was considered "the best" at the time. And ambulances owned by Federal agencies (such as the military) had to meet the Federal specs - with some exceptions, they still do.
But yes, if you wanted to buy an ambulance not using Federal money, and there was no other requirement that it meet the Federal specs, you have always been allowed to paint it whatever color you wanted.